If I integrate $\int e^{iz}\,dz$ for z complex, along the positive real line, then is the imaginary part of the integral $i\int \sin(x)\,dx$ automatically equal to zero (integration only along the real line)?
I know that integrating a real function over a real interval can only ever result in a real number answer, but in this case, the integrand is a complex number -- Euler's formula. So, I am a little confused.
However, I have a feeling that the imaginary part of the integral must be zero, a proof that I am working on depends on this to happen. So, I just wanted to confirm.
Thanks